The phrase of the day is, “Quarterback Sneak”. It is a running play when the QB takes the snap from under center and moves forward behind the Guard/Center/Guard trio finding a hole if possible. It’s usually run in short yardage situations.
There are times that the offense loads up with extra lineman/TEs to move the pile, in an attempt to get the needed yardage. There are times when a QB can see something in the defensive front, and go up to the center with a 3WR or 4WR set and snap the ball to get the needed yardage. Unfortunately, Tom Brady has mastered this because he can catch the defense out of position and easily picks up the first down.
The Running Back/Full Back (depending on the formation) often slam into the back of the QB trying to help push/move him further down the field. It’s a dangerous technique, but at times it’s likely the player simply reacting to the need of the situation.